Nonlinear PDEs and measure-valued branching type processes

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• 作者：Lucian Beznea ; ^a ; ^{lucian.beznea@imar.ro"" rel=""nofollow} ; Andrei-George Oprina^a ; ^{oandrei22@yahoo.com"" rel=""nofollow}
• 关键词：Nonlinear PDE ; Discrete branching ; Continuous branching ; Measure-valued Markov process ; Continuous additive functional ; Branching kernel ; Nonlinear Dirichlet problem
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2011
• 出版时间：1 December 2011
• 年：2011
• 卷：384
• 期：1
• 页码：16-32
• 全文大小：282 K

文摘

We deal with the probabilistic approach to a nonlinear operator Λ of the form , in connection with the works of M. Nagasawa, N. Ikeda, S. Watanabe, and M.L. Silverstein on the discrete branching processes. Instead of the Laplace operator we may consider the generator of a right (Markov) process, called base process, with a general (not necessarily locally compact) state space. It turns out that solutions of the nonlinear equation Λu=0 are produced by the harmonic functions with respect to the (linear) generator of a discrete branching type process. The consideration of the general state space allows to take as base process a measure-valued superprocess (in the sense of E.B. Dynkin). The probabilistic counterpart is a Markov process which is a combination between a continuous branching process (e.g., associated with a nonlinear operator of the form Δu−u^α, 1c;α2) and a discrete branching type one, on a space of configurations of finite measures. Our approach uses probabilistic and analytic potential theoretical tools, like the potential kernel of a continuous additive functional and the subordination operators.